Question

A source emits a frequency of 232Hz. An observer is moving away from the source with velocity 20$m{{s}^{-1}}$ and the source is moving away from the observer with a velocity 10$m{{s}^{-1}}$ along the straight line. The wind is blowing in a direction from the observer to the source with a velocity 5$m{{s}^{-1}}$. The apparent frequency heard by the observer is (velocity of sound = 343$m{{s}^{-1}}$)
A. 244 Hz
B. 242 Hz
C. 212 Hz
D. 237 Hz

Answer 1

Hint: When there is relative motion between the source of sound and the observer, the frequency of the sound heard by the observer is different from the actual frequency of the sound. Use the formula for the apparent frequency when the source of sound and observer are moving away from each other and find the apparent frequency.

Formula used: ${{f}^{'}}=f\left( \dfrac{v-{{v}_{o}}-w}{v+{{v}_{s}}-w} \right)$

Complete step by step answer:
It is given that a source of sound is moving away from the observer and the observer is moving away from the source. The frequency of the sound emitted by the source is 232Hz and the source is moving with a speed of 10$m{{s}^{-1}}$ and the observer is moving a speed of 20$m{{s}^{-1}}$. It is also given that the speed of sound in this medium is 343 m/s.
When there is relative motion between the source of sound and the observer, the frequency of the sound heard by the observer is different from the actual frequency of the sound (i.e. the frequency of the sound that is heard when the observer and the source are at rest).
When the observer and the source are moving away from each other, the apparent frequency is given as ${{f}^{'}}=f\left( \dfrac{v-{{v}_{o}}}{v+{{v}_{s}}} \right)$ …. (i),
where f is the actual frequency, v is the speed of the sound, ${{v}_{s}}$ is the speed of the source and ${{v}_{o}}$ is the speed of the observer.
And when wind blows from the observer to the source with a speed w, the apparent frequency becomes ${{f}^{'}}=f\left( \dfrac{v-{{v}_{o}}-w}{v+{{v}_{s}}-w} \right)$ …. (i).
In this case, f = 232 Hz, $v=343m{{s}^{-1}}$, ${{v}_{s}}=10m{{s}^{-1}}$, ${{v}_{o}}=20m{{s}^{-1}}$ and $w=5m{{s}^{-1}}$
Substitute the values in (i).
$\Rightarrow {{f}^{'}}=232\left( \dfrac{343-20-5}{343+10-5} \right)=232\left( \dfrac{318}{348} \right)=212Hz$

So, the correct answer is “Option C”.

Note: Remember that when the source moves towards a stationary observer, frequency of the sound increases. When the source moves away from a stationary observer, frequency of the sound decreases.